4.3 - Properties of Logarithms

Change of Base Formula

One dilemma is that your calculator only has logarithms for two bases on it. Base 10 (log) and base e (ln). What is to happen if you want to know the logarithm for some other base? Are you out of luck?

No. There is a change of base formula for converting between different bases. To find the log base a, where a is presumably some number other than 10 or e, otherwise you would just use the calculator,

Take the log of the argument divided by the log of the base.

log_a x = ( log_b x ) / ( log_b a )

There is no need that either base 10 or base e be used, but since those are the two you have on your calculator, those are probably the two that you're going to use the most. I prefer the natural log (ln is only 2 letters while log is 3, plus there's the extra benefit that I know about from calculus). The base that you use doesn't matter, only that you use the same base for both the numerator and the denominator.

log_a x = ( log x ) / ( log a ) = ( ln x ) / ( ln a )

Example: log₃ 7 = ( ln 7 ) / ( ln 3 )

Logarithms are Exponents

Remember that logarithms are exponents, so the properties of exponents are the properties of logarithms.

Multiplication

What is the rule when you multiply two values with the same base together (x² * x³)? The rule is that you keep the base and add the exponents. Well, remember that logarithms are exponents, and when you multiply, you're going to add the logarithms.

The log of a product is the sum of the logs.

log_a xy = log_a x + log_a y

Division

The rule when you divide two values with the same base is to subtract the exponents. Therefore, the rule for division is to subtract the logarithms.

The log of a quotient is the difference of the logs.

log_a (x/y) = log_a x - log_a y

Raising to a Power

When you raise a quantity to a power, the rule is that you multiply the exponents together. In this case, one of the exponents will be the log, and the other exponent will be the power you're raising the quantity to.

The exponent on the argument is the coefficient of the log.

log_a x^r = r * log_a x

Melodic Mathematics

Some of the statements above are very melodious. That is, they sound good. It may help you to memorize the melodic mathematics, rather than the formula.

• The log of a product is the sum of the logs
• The sum of the logs is the log of the products
• The log of a quotient is the difference of the logs
• The difference of the logs is the log of the quotient
• The exponent on the argument is the coefficient of the log
• The coefficient of the log is the exponent on the argument

Okay, so the last two aren't so melodic.

Common Mistakes

I almost hesitate to put this section in here. It seems when I try to point out a mistake that people are going to make, that more people make it.

• The log of a sum is NOT the sum of the logs. The sum of the logs is the log of the product. The log of a sum cannot be simplified.
  log_a (x + y) ≠ log_a x + log_a y
• The log of a difference is NOT the difference of the logs. The difference of the logs is the log of the quotient. The log of a difference cannot be simplified.
  log_a (x - y) ≠ log_a x - log_a y
• An exponent on the log is NOT the coefficient of the log. Only when the argument is raised to a power can the exponent be turned into the coefficient. When the entire logarithm is raised to a power, then it can not be simplified.
  (log_a x)^r ≠ r * log_a x
• The log of a quotient is not the quotient of the logs. The quotient of the logs is from the change of base formula. The log of a quotient is the difference of the logs.
  log_a (x / y) ≠ ( log_a x ) / ( log_a y )